Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $r = \dfrac{2p^2 + 14p - 16}{3p^3 + 9p^2 - 120p} \times \dfrac{-4p^2 + 20p}{-5p + 10} $
Explanation: First factor out any common factors. $r = \dfrac{2(p^2 + 7p - 8)}{3p(p^2 + 3p - 40)} \times \dfrac{-4p(p - 5)}{-5(p - 2)} $ Then factor the quadratic expressions. $r = \dfrac {2(p + 8)(p - 1)} {3p(p + 8)(p - 5)} \times \dfrac {-4p(p - 5)} {-5(p - 2)} $ Then multiply the two numerators and multiply the two denominators. $r = \dfrac { 2(p + 8)(p - 1) \times -4p(p - 5)} { 3p(p + 8)(p - 5) \times -5(p - 2)} $ $r = \dfrac {-8p(p + 8)(p - 1)(p - 5)} {-15p(p + 8)(p - 5)(p - 2)} $ Notice that $(p + 8)$ and $(p - 5)$ appear in both the numerator and denominator so we can cancel them. $r = \dfrac {-8p\cancel{(p + 8)}(p - 1)(p - 5)} {-15p\cancel{(p + 8)}(p - 5)(p - 2)} $ We are dividing by $p + 8$ , so $p + 8 \neq 0$ Therefore, $p \neq -8$ $r = \dfrac {-8p\cancel{(p + 8)}(p - 1)\cancel{(p - 5)}} {-15p\cancel{(p + 8)}\cancel{(p - 5)}(p - 2)} $ We are dividing by $p - 5$ , so $p - 5 \neq 0$ Therefore, $p \neq 5$ $r = \dfrac {-8p(p - 1)} {-15p(p - 2)} $ $ r = \dfrac{8(p - 1)}{15(p - 2)}; p \neq -8; p \neq 5 $